Subjective Type

How do you find the factorial of negative numbers ?

Solution

The factorial function in the traditional sense takes only non-negative integers as the domain, with the convention that $$0! = 1$$
However the function can be extended to the range of all real numbers using the Gamma function,
$$\Gamma (z) = \displaystyle \int_{0}^{\infty} t^{z - 1}e^{-t}dt$$
which is what you have graphed. The gamma function is not the same as the factorial function, however it does have the property that for positive numbers that:
$$n! = \Gamma (n + 1)$$
Using the gamma function we can therefore put a meaning to fractional and negative values, so for example:
$$\Gamma \left ( \dfrac{1}{2} \right ) = \left ( -\dfrac{1}{2} \right )! = \sqrt{\pi}$$

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